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Transmission Line Equations

Transmission lines take on many forms in order to accommodate particular applications. All rely on the same basic components - two or more conductors separated by a dielectric (insulator). The physical configuration and properties of all the components determines the characteristic impedance, distortion, transmission speed, and loss.

The following formulas are presented in a compact text format that can be copied and pasted into a spreadsheet or other application.

For the following equations, ε is the dielectric constant (ε = 1 for air)





Two Conductors in Parallel (Unbalanced)
Above Ground Plane


For D << d, h

Z0= (69/ε½) log10{(4h/d)[1+(2h/D)2]}

Parallel conductors above a ground plane - RF Cafe
Single Conductor Above Ground Plane
 

For d << h

Z0= (138/ε½) log10(4h/d)

Single conductor above a ground plane - RF Cafe
Two Conductors in Parallel (Balanced)
Above Ground Plane


For D << d, h1, h2

Z0= (276/ε½) log10{(2D/d)[1+(D/2h)2]}

Parallel conductors above a ground plane - RF Cafe
Two Conductors in Parallel (Balanced)
Different Heights Above Ground Plane


For D << d, h1, h2

Z0= (276/ε½)log10{(2D/d)[1+(D2/4h1h2)]}

Parallel conductors different heights above a ground plane - RF Cafe
Single Conductor Between
Parallel Ground Planes


For d/h << 0.75

Z0= (138/ε½) log10(4h/πd)

Single conductor between parallel ground planes - RF Cafe

Two Conductors in Parallel (Balanced)
Between Parallel Ground Planes


For d << D, h

Z0= (276/ε½) log10{[4h tanh(πD/2h)]/πd}

Two conductors between parallel ground planes - RF Cafe
Balanced Conductors Between
Parallel Ground Planes


For d << h

Z0= (276/ε½) log10(2h/πd)

Balanced conductors between parallel ground planes - RF Cafe

Two Conductors in Parallel (Balanced)
of Unequal Diameters





Z0= (60/ε½) cosh-1 (N)

N = ½[(4D2/d1d2) - (d1/d2) - (d2/d1)]

Parallel conductors - unequal diameters - RF Cafe
Balanced 4-Wire Array

For d << D1, D2

Z0= (138/ε½) log10{(2D2/d)[1+(D2/D1)2]}

Balanced 4-wire array - RF Cafe
Two Conductors
in Open Air


Z0= 276 log10(2D/d)

Two conductors in open air - RF Cafe
5-Wire Array

For d << D

Z0= (173/ε½) log10(D/0.933d)

5-wire array - RF Cafe
Single Conductor in
Square Conductive Enclosure


For d << D

Z0≈ [138 log10(ρ) +6.48-2.34A-0.48B-0.12C]/ε½

A = (1+0.405ρ-4)/(1-0.405ρ-4)

B = (1+0.163ρ-8)/(1-0.163ρ-8)

C = (1+0.067ρ-12)/(1-0.067ρ-12)

ρ= D/d

Single conductor in square conducting enclosure - RF Cafe

Air Coaxial Cable with
Dielectric Supporting Wedge


For d << D

Z0≈ [138 log10(D/d)]/[1+(ε-1)(θ/360)]½)

ε = wedge dielectric constant

θ= wedge angle in degrees

Air coaxial cable with dielectric supporting wedge - RF Cafe
Two Conductors Inside Shield
(sheath return)


For d << D, h

Z0= (69/ε½) log10[(ν/2σ2)(1-σ4)]

ν = h/d       σ = h/D

Twin conductors inside shield - RF Cafe

Balanced Shielded Line

For D>>d, h>>d

Z0= (276/ε½) log10{2ν[(1-σ2)/(1+σ2)]}

ν = h/d       σ = h/D

Balanced shielded line equation - RF Cafe
 
Two Conductors in Parallel (Unbalanced)
Inside Rectangular Enclosure


For d << D, h, w

                          ∞
Z0= (276/ε½) {log10[(4h tanh(πD/2h)/πd)- ∑ log10[(1+μm2)/(1-νm2)]}
                            m=1

μm=sinh(πD/2h)/cosh(mπw/2h)

νm=sinh(πD/2h)/sinh(mπw/2h)

Balanced 2-conductor line inside rectangular enclosure - RF Cafe


Equations appear in "Reference Data for Engineers," Sams Publishing 1993
ERZIA (RF amplifiers, wireless, communications) - RF Cafe ConductRF D38999 RF Cables - RF Cafe
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Copyright: 1996 - 2018
Webmaster:
    Kirt Blattenberger,
    BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

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